11.4 angle measures and
segment lengths
SWBAT…
• Find the measure of angles formed
by chords, secants and tangents
• Find the lengths of segments
associated with circles
Lines Intersecting Inside or
Outside a Circle
• If two lines intersect a circle, there
are three (3) places where the lines
can intersect.
on the circle
Inside the circle
Outside the circle
Lines Intersecting
• You know how to find angle and arc
measures when lines intersect
ON THE CIRCLE.
• You can use the following theorems
to find the measures when the lines
intersect
INSIDE or OUTSIDE the circle.
D
Theorem
• If two chords intersect
in the interior of a
circle, then the
measure of each angle
is one half the sum of
the measures of the
arcs intercepted by the
angle and its vertical
angle.
1
A
C
2
B
 
 
m1 = ½ m CD + m
AB
m2 = ½ m
AD
BC
+m
Theorem
B
A
• If a tangent and a
secant, two tangents
or two secants
intercept in the
EXTERIOR of a circle,
then the measure of
the angle formed is
one half the difference
of the measures of the
intercepted arcs.
1
C
 
m1 = ½ m( BC - m
AC )
Theorem
• If a tangent and a
secant, two tangents
or two secants
intercept in the
EXTERIOR of a circle,
then the measure of
the angle formed is
one half the difference
of the measures of the
intercepted arcs.
P
2
R
Q
 
m2 = ½ m( PQR - m PR )
Theorem
X
• If a tangent and a
secant, two tangents 3
or two secants
intercept in the
EXTERIOR of a circle,
then the measure of
the angle formed is
one half the difference
of the measures of the
intercepted arcs.
W
Z
Y
 
m3 = ½ m( XY - mWZ )
Ex. 3: Finding the Measure of an
Angle Formed by Two Chords 106°
P
• Find the value of x
S
Q
x°
R
 
• Solution:
x° = ½ (mPS +m RQ
x° = ½ (106° + 174°)
x = 140
174°
Apply Theorem 10.13
Substitute values
Simplify
E
Ex. 4: Using Theorem 10.14
200°
• Find the value of x
Solution:
 
mGHF = ½ m(EDG - m
GF
72° = ½ (200° - x°)
144 = 200 - x°
- 56 = -x
56 = x
D
F
x°
G
) Apply Theorem 10.14
H
72°
Substitute values.
Multiply each side by 2.
Subtract 200 from both sides.
Divide by -1 to eliminate negatives.
Ex. 4: Using Theorem 10.14
 
M
Because MN and MLN make a
whole circle, m MLN =360°-92°=268°
L
• Find the value of x
Solution:
N
 
mGHF = ½ m(MLN - m MN ) Apply Theorem 10.14
= ½ (268 - 92)
= ½ (176)
= 88
92°
Substitute values.
Subtract
Multiply
x°
P
Finding the Lengths of Chords
• When two chords intersect in the
interior of a circle, each chord is
divided into two segments which are
called segments of a chord. The
following theorem gives a
relationship between the lengths of
the four segments that are formed.
B
Theorem
• If two chords
intersect in the
interior of a circle,
then the product of
the lengths of the
segments of one
chord is equal to
the product of the
lengths of the
segments of the
other chord.
C
E
D
A
EA • EB = EC • ED
Ex. 1: Finding Segment
Lengths
• Chords ST and PQ
intersect inside the
circle. Find the
value of x.
Q
S
9
R
X
6
T
RQ • RP = RS • RT
Use Theorem 10.15
9•x=3•6
Substitute values.
Simplify.
9x = 18
x=2
3
Divide each side by 9.
P
Using Segments of Tangents
and Secants
• In the figure shown,
PS is called a
P
tangent segment
because it is tangent
to the circle at an
end point. Similarly,
PR is a secant
segment and PQ is
the external segment
of PR.
R
Q
S
Theorem
• If two secant
A
segments share the
same endpoint outside E
C
a circle, then the
product of the length
of one secant
segment and the
EA • EB = EC • ED
length of its external
segment equals the
product of the length
of the other secant
segment and the
length of its external
segment.
B
D
Theorem
A
• If a secant segment
and a tangent
E
segment share an
C
endpoint outside a
circle, then the
product of the length
of the secant segment (EA)2 = EC • ED
and the length of its
external segment
equal the square of
the length of the
tangent segment.
D
Ex. 2: Finding Segment
Lengths
P
Q
11
9
• Find the value of x.
R
10
S
x
RP • RQ = RS • RT
Use Theorem
9•(11 + 9)=10•(x + 10) Substitute values.
Simplify.
180 = 10x + 100
Subtract 100 from each side.
80 = 10x
Divide each side by 10.
8 =x
T
Ex. 3: Estimating the radius of
a circle
• Aquarium Tank.
You are standing
at point C, about 8
feet from a circular
aquarium tank.
The distance from
you to a point of
tangency is about
20 feet. Estimate
the radius of the
tank.
(CB)2 = CE • CD
(20)2  8 • (2r + 8)
400  16r + 64
336  16r
21  r
Use Theorem 10.17
Substitute values.
Simplify.
Subtract 64 from each side.
Divide each side by 16.
So, the radius of the tank is about 21 feet.
(BA)2 = BC • BD
(5)2 = x • (x + 4)
25 = x2 + 4x
0 = x2 + 4x - 25
Use Theorem 10.17
Substitute values.
Simplify.
Write in standard form.
 4  42  4(1)(25) Use Quadratic Formula.
2
x=
Simplify.
x=
 2  29
Use the positive solution because lengths cannot be
negative. So, x = -2 + 29  3.39.
You try…
• Page 611 #’s 1-16
Homework…
• Page 612 #’s 20 – 25, 28